Question

How do you use the quotient rule to differentiate `1 / (1 + x²)`?

Answer 1

The quotient rule states that the derivative of some function that's expressed as a quotient of two other functions, such as if `f(x)=(g(x))/(h(x))`, then the derivative of `f` is given through:

`f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))^2`

For `f(x)=1/(1+x^2)`, we see that `g(x)=1` and `h(x)=1+x^2`.

We then see that `g'(x)=0` and `h'(x)=2x`. Plugging these in gives:

`f'(x)=(0(1+x^2)-1(2x))/(1+x^2)^2`

`f'(x)=(-2x)/(1+x^2)^2`

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Footnote

If you've learned the chain rule, it's easier to do this by rewriting the function as `(1+x^2)^-1` and then seeing that the derivative is `-(1+x^2)^-2d/dx(x^2)`, identical to what we found.